Water flows through a horizontal tube as shown in figure (13-E9).
Question: Water flows through a horizontal tube as shown in figure (13-E9). If the difference of heights of water column in the vertical tubes is $2 \mathrm{~cm}$, and the areas of cross-section at $A$ and $B$ are $4 \mathrm{~cm}^{2}$ and $2 \mathrm{~cm}^{2}$ respectively, find the rate of flow of water across any section. Solution:...
Read More →Evaluate the following integrals:
Question: Evaluate the following integrals: $\int \frac{d x}{(1-\tan x)}$ Solution:...
Read More →Water flows through a tube shown in figure (13-E8).
Question: Water flows through a tube shown in figure (13-E8). The areas of cross-section at $\mathrm{A}$ and $\mathrm{B}$ are $1 \mathrm{~cm}^{2}$ and0.5 $\mathrm{cm}^{2}$ respectively. The height difference between $A$ and $B$ is $5 \mathrm{~cm}$. If the speed of water at $A$ is $10 \mathrm{~cm} / \mathrm{s}$ fid (a) the speed at $B$ (b) the difference in pressures at $\mathrm{A}$ and $\mathrm{B}$. Solution:...
Read More →Suppose the tube in the previous problem is kept vertical with B upward.
Question: Suppose the tube in the previous problem is kept vertical with B upward. Water enters through B at the rate of $1 \mathrm{~cm} 3 / \mathrm{s}$. Repeat parts (a), (b) and (c). Note that the speed decreases as the water falls down. Solution:...
Read More →Evaluate the following integrals:
Question: Evaluate the following integrals: $\int \frac{\sin x}{(\sin x-\cos x)} d x$ Solution:...
Read More →Supposed the tube in the previous problem is kept vertical with
Question: Supposed the tube in the previous problem is kept vertical with A upward but the other conditions remain the same. The separation between the cross-section at $\mathrm{A}$ and $\mathrm{B}$ is $15 / 16 \mathrm{~cm}$. Repeat parts $(\mathrm{a})$, (b) and (c) of the previous problem. Take $\mathrm{g}=$ $10 \mathrm{~m} / \mathrm{s}^{2}$. Solution:...
Read More →Water flows through a horizontal tube of variable cross-section (figure 13-E7).
Question: Water flows through a horizontal tube of variable cross-section (figure 13-E7). The area of cross-section at A and B are 4 $\mathrm{mm}^{2}$ and $2 \mathrm{~mm}^{2}$ respectively. If $1 \mathrm{cc}$ of water enters per second through $\mathrm{A}$, find (a) the speed of water at A, (b) the speed of water at $B$ and (c) the pressure difference $P_{A}-P_{B}$. Solution:...
Read More →Evaluate the following integrals:
Question: Evaluate the following integrals: $\int(2 x+4) \sqrt{x^{2}+4 x+3} d x$ Solution:...
Read More →At Deoprayag (Garhwal, UP) river Alaknanda mixes with the river Bhagirathi and
Question: At Deoprayag (Garhwal, UP) river Alaknanda mixes with the river Bhagirathi and becomes river Ganga. Suppose Alaknanda has a width of $12 \mathrm{~m}$, Bhagirathi has a width of $8 \mathrm{~m}$ and Ganga has a width of $16 \mathrm{~m}$. Assume that the depth of water is same in the three rivers. Let the average speed of water in Alaknanda be $20 \mathrm{~km} / \mathrm{h}$ and in Bhagirathi be $16 \mathrm{~km} / \mathrm{h}$. Find the average speed of water in the river Ganga. Solution:...
Read More →Evaluate the following integrals:
Question: Evaluate the following integrals: $\int x^{2} \sin x^{3} d x$ Solution:...
Read More →A U-tube containing a liquid is accelerated horizontally with
Question: A U-tube containing a liquid is accelerated horizontally with a constant acceleration $\mathrm{a}_{0}$. If the separation between the vertical limbs is I, find the difference in the heights of the liquid in the two arms. Solution:...
Read More →A cube of ice of edge 4 cm is placed in an empty cylindrical glass of inner diameter 6 cm.
Question: A cube of ice of edge $4 \mathrm{~cm}$ is placed in an empty cylindrical glass of inner diameter $6 \mathrm{~cm}$. Assume that the ice melts uniformly from each side so that it always retains its cubical shape. Remembering that ice is lighter than water, find the length of the edge of the ice cube at the instant it just contact with the bottom of the glass. Solution:...
Read More →Evaluate the following integrals:
Question: Evaluate the following integrals: $\int \frac{d x}{x \cos ^{2}(1+\log x)}$ Solution:...
Read More →A wooden block of mass 0.5 kg and density 800 kg/m3is fastened to the free end
Question: A wooden block of mass $0.5 \mathrm{~kg}$ and density $800 \mathrm{~kg} / \mathrm{m}^{3}$ is fastened to the free end of a vertical spring of spring constant 50 $\mathrm{N} / \mathrm{m}$ fixed at the bottom. If the entire system is completely immersed in water, find (a) the elongation (or compression) of the spring in equilibrium and (b) the time-period of vertical oscillations of the block when it is slightly depressed and released. Solution:...
Read More →Evaluate the following integrals:
Question: Evaluate the following integrals: $\int x \sqrt{3 x-2} d x$ Solution:...
Read More →A cylindrical object of outer diameter 10 cm,
Question: A cylindrical object of outer diameter $10 \mathrm{~cm}$, height $20 \mathrm{~cm}$ and density $8000 \mathrm{~kg} / \mathrm{m}^{3}$ is supported by a vertical spring and is half dipped in water as shown in figure(13-E6). (a) Find the elongation of the spring in equilibrium condition. (b) If the object is slightly depressed and released, find the time period of resulting oscillations of the object. The spring constant $=500 \mathrm{~N} / \mathrm{m}$. Solution:...
Read More →Evaluate the following integrals:
Question: Evaluate the following integrals: $\int x \sqrt{x^{2}-1} d x$ Solution:...
Read More →A cylindrical object of outer diameter 20 cm and mass 2 kg floats in water with its axis vertical.
Question: A cylindrical object of outer diameter $20 \mathrm{~cm}$ and mass $2 \mathrm{~kg}$ floats in water with its axis vertical. If it is slightly depressed and then released, find the time period of the resulting simple harmonic motion of the object. Solution:...
Read More →Evaluate the following integrals:
Question: Evaluate the following integrals: $\int(1-x) \sqrt{1+x} d x$ Solution:...
Read More →Find the ratio of the weights, as measured by a spring balance,
Question: Find the ratio of the weights, as measured by a spring balance, of a $1 \mathrm{~kg}$ block of iron and a $1 \mathrm{~kg}$ block of wood. Density of iron $=7800 \mathrm{~kg} / \mathrm{m}^{3}$, density of wood $=800 \mathrm{~kg} / \mathrm{m}^{3}$ and density of air $=1.293 \mathrm{~kg} / \mathrm{m}^{3}$. Solution:...
Read More →A solid sphere of radius 5 cm floats in water.
Question: A solid sphere of radius $5 \mathrm{~cm}$ floats in water. If a maximum load of $0.1 \mathrm{~kg}$ can be put on it without wetting the load, find the specific gravity of the material of the sphere. Solution:...
Read More →Evaluate the following integrals:
Question: Evaluate the following integrals: $\int \mathrm{x} \sqrt{-1} \mathrm{dx}$ Solution:...
Read More →A hollow spherical body of inner and outer radii
Question: A hollow spherical body of inner and outer radii $6 \mathrm{~cm}$ and $8 \mathrm{~cm}$ respectively floats half submerged in water. Find the density of the material of the sphere. Solution:...
Read More →Evaluate the following integrals:
Question: Evaluate the following integrals: $\int \frac{1}{x \sqrt{x^{4}-1}} d x$ Solution:...
Read More →Evaluate the following integrals:
Question: Evaluate the following integrals: $\int \frac{x}{\sqrt{1+x}} d x$ Solution:...
Read More →